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!pip install numpy pandas matplotlib seaborn statsmodels scipy copulalib pandas_montecarlo
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import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
import pandas_montecarlo
import copy
import statsmodels.api as sm # statistical model library
import scipy
from copulalib.copulalib import Copula
%matplotlib inline
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files = {"amd-usd": r"https://bit-rapid.com/ds_homework/data/amd.csv",
"AT&S-usd": r"https://bit-rapid.com/ds_homework/data/AT&S_bereinigt.csv",
"btc-eur": r"https://bit-rapid.com/ds_homework/data/btc-eur-max.csv",
"btc-usd": r"https://bit-rapid.com/ds_homework/data/btc-usd-max.csv",
"crude-oil-prices-europe-usd": r"https://bit-rapid.com/ds_homework/data/crude_oil_prices_brent_europe.csv",
"dax-eur": r"https://bit-rapid.com/ds_homework/data/DAX.csv",
#"eur-usd": r"https://bit-rapid.com/ds_homework/data/eur-usd.csv",
"gold-usd": r"https://bit-rapid.com/ds_homework/data/gold.csv",
"intel-usd": r"https://bit-rapid.com/ds_homework/data/Intel_bereinigt.csv",
"ltc-eur": r"https://bit-rapid.com/ds_homework/data/ltc_eur_max.csv",
"ltc-usd": r"https://bit-rapid.com/ds_homework/data/ltc_usd_max.csv",
"nikkei": r"https://bit-rapid.com/ds_homework/data/Nikkei 225.csv",
"sp500": r"https://bit-rapid.com/ds_homework/data/sp500.csv",
"ripple-usd": r"https://bit-rapid.com/ds_homework/data/xrp_usd.csv",
"nvidia": r"https://bit-rapid.com/ds_homework/data/nvidia.csv",
"ripple-eur": r"https://bit-rapid.com/ds_homework/data/xrp_eur.csv"}
data = {}
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dates = pd.date_range('01-01-2014', '10-31-2017')
for key, value in files.items():
data[key] = pd.read_csv(files[key], sep=";", decimal=",")
data[key]['date'] = pd.to_datetime(data[key]['date'], format='%Y-%m-%d')
data[key] = data[key].sort_values(by=['date'], ascending=[True])
data[key].set_index('date', inplace=True)
data[key] = data[key].resample('D').ffill().reset_index()
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def plot_datePrice(string):
price = data[string]['price']
date = data[string]['date']
plt.plot(date, price)
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def findCorrelation(string, string1):
return scipy.stats.spearmanr(data[string]['price'], data[string1]['price'])[0][1]
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prices = pd.DataFrame()
for key, value in data.items():
prices = pd.concat([prices, data[key]['price'].rename('price ' + key)], axis=1)
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
f, ax = plt.subplots(figsize=(10, 8))
corr = prices.corr(method='spearman')
sns.heatmap(corr, mask=np.zeros_like(corr, dtype=np.bool), cmap=sns.diverging_palette(220, 10, as_cmap=True),
square=True, ax=ax)
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Heat Map Description¶
This Heat Map shows the correlation between the historical prices of the Cryptocurrencies Bitcoin (BTC), Litecoin (LTC), Ripple (XRP) and the chosen stock data in the time from 01/01/2014 - 31/10/2017.
This first Heat Map illustrates all the stock exchange data that we initially expected to have a high correlation to at least one of the Cryptocurrencies. After preparing the data and visualizing the correlations in this Heat Map, we have noticed that some of the chosen data does not have a high correlation or even a negative correlation to the Cryptocurrencies. For this reason we have taken the following data out of the scope:
- Gold price
- Oil price
- AT&S Stock Exchange
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
prices = pd.DataFrame()
for key, value in data.items():
if key not in ['gold-usd', 'crude-oil-prices-europe-usd', 'AT&S-usd']:
prices = pd.concat([prices, data[key]['price'].rename('price ' + key)], axis=1)
f, ax = plt.subplots(figsize=(10, 8))
corr = prices.corr(method='spearman')
sns.heatmap(corr, mask=np.zeros_like(corr, dtype=np.bool), cmap=sns.diverging_palette(220, 10, as_cmap=True),
square=True, ax=ax)
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This improved Heat Map now shows only positive correlated data. For further improvement, we have in a next step also removed the stock exchange data of Nikkei and Intel.
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
prices = pd.DataFrame()
for key, value in data.items():
if key not in ['gold-usd', 'AT&S-usd', 'nikkei', 'intel-usd', 'ripple-eur', 'btc-eur', 'ltc-eur']:
prices = pd.concat([prices, data[key]['price'].rename('price ' + key)], axis=1)
f, ax = plt.subplots(figsize=(10, 8))
corr = prices.corr(method='spearman')
sns.heatmap(corr, mask=np.zeros_like(corr, dtype=np.bool), cmap="Reds", square=True, ax=ax)
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By this further optimization we have been able to illustrate a Heat Map which shows only data with a correlation greater than 0,5. A high correlation (between 0,5 and 0,8) can be observed for example between:
- BTC and AMD
- BTC and DAX
- BTC and SP500
- LTC and AMD
- LTC and DAX
- LTC and SP500
- LTC and NVIDIA
- XRP and AMD
- XRP and DAX
- XRP and SP500
While a very high correlation (between 0,8 and 1,0) can be seen between:
- BTC and LTC
- BTC and XRP
- BTC and NVIDIA
- LTC and BTC
- LTC and XRP
- XRP and BTC
- XRP and LTC
- XRP and NVIDIA
This comparison shows, that all three Cryptocurrencies show a quite high correlation to AMD, DAX, SP500 and NVIDIA. Nevertheless it cannot be stated if the currencies are directly correlated to the stock exchange rates or if the correlation depends on other variables.
Price/percentage changes Plots¶
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
(prices['price ltc-usd'].pct_change()*100).plot(kind="kde")
print("Jarque–Bera test", scipy.stats.jarque_bera(prices['price ltc-usd'].pct_change()[1:]))
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
(prices['price btc-usd'].pct_change()*100).plot(kind="kde")
print("Jarque–Bera test", scipy.stats.jarque_bera(prices['price btc-usd'].pct_change()[1:]))
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
(prices['price ripple-usd'].pct_change()*100).plot(kind="kde")
print("Jarque–Bera test", scipy.stats.jarque_bera(prices['price btc-usd'].pct_change()[1:]))
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
plt.plot(prices['price nvidia'].pct_change()*100, c="red")
plt.plot(prices['price btc-usd'].pct_change()*100, c="blue")
plt.show()
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
plt.plot(prices['price amd-usd'].pct_change()*100, c="red")
plt.plot(prices['price btc-usd'].pct_change()*100, c="blue")
plt.show()
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
plt.plot(prices['price nvidia'][-60:-1].pct_change(), c="red")
plt.plot(prices['price btc-usd'][-60:-1].pct_change(), c="blue")
plt.show()
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
print(prices['price nvidia'].pct_change().describe())
print(prices['price btc-usd'].pct_change().describe())
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
data['nvidia']['price'].pct_change().plot(kind="hist")
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
data['btc-usd']['price'].pct_change().plot(kind="hist")
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Monte Carlo Simulations¶
By using Monte Carlo Simulations, we have simulated 1000 possible results to show how the exchange rates/stock prices could continue during the next 30 days. After comparing the results of the Monte Carlo Simulations on the normal distributed data and the mean with the real price data, we have noticed that the results of the Monte Carlo Simulations are generally 'too optimistic'.
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
import numpy as np
import math
import matplotlib.pyplot as plt
from scipy.stats import norm
def monte_carlo(t, name):
days = len(data[name])
cagr = (((data[name]['price'].iloc[-1] / data[name]['price'].iloc[0])) ** (365.0/days)) - 1
mu = cagr
returns = prices['price ' + name].pct_change()
vol = prices['price ' + name].pct_change().std()*math.sqrt(365)
T = t
daily_returns=np.random.normal((mu/T),vol/math.sqrt(T),T)+1
S = data[name]['price'].iloc[-1]
#set starting price and create price series generated by above random daily returns
price_list = [S]
for x in daily_returns[0:T]:
price_list.append(price_list[-1]*x)
print('-----------------', name , '-----------------')
print ('CAGR =',str(round(cagr,4)*100)+"%")
print ("Annual Volatility =",str(round(vol,4)*100)+"%")
#Generate Plots - price series and histogram of daily returns
plt.plot(price_list)
plt.show()
plt.hist(daily_returns-1, 100) #Note that we run the line plot and histogram separately, not simultaneously.
plt.show()
result = []
for i in range(1000):
daily_returns=np.random.normal(mu/T,vol/math.sqrt(T),T)+1
price_list = [S]
for x in daily_returns[0:T]:
price_list.append(price_list[-1]*x)
plt.plot(price_list)
result.append(price_list[-1])
plt.show()
plt.axvline(np.percentile(result,25), color='r', linestyle='dashed', linewidth=2)
plt.axvline(np.percentile(result,50), color='r', linestyle='dashed', linewidth=2)
plt.axvline(np.percentile(result,75), color='r', linestyle='dashed', linewidth=2)
plt.hist(result,bins=50)
plt.show()
print("Mean =", np.mean(result[0:30]))
for key, value in data.items():
if key in ['btc-usd', 'ltc-usd', 'ripple-usd']:
pass
monte_carlo(30, 'btc-usd')
monte_carlo(30, 'ltc-usd')
monte_carlo(30, 'ripple-usd')
monte_carlo(30, 'nvidia')
monte_carlo(30, 'sp500')
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np.std(prices['price amd-usd'].pct_change())
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def monte_carlo_multivariate(price1, price2, days):
means = [np.mean(price1.pct_change().loc[1:]), np.mean(price2.pct_change().loc[1:])]
cov = np.cov(price1.pct_change().loc[1:], price2.pct_change().loc[1:])
normal_multi = np.random.multivariate_normal(means, cov, days)
for i in range(len(normal_multi)):
normal_multi[i-1] = normal_multi[i-1]+1
A = [0]*days
A[0] = [price1.loc[1399], price2.loc[1399]]
#print(normal_multi[0])
#print(A[0])
for i in range(1,days):
A[i] = A[i-1]*normal_multi[i-1]
return A
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
# index must be either 1 or 0, 1 for the second value, 0 for the first
# A is the return of monte_carlo_multivariate
def fan_chart(price1, price2, days, iterations=1000):
carlos = []
for i in range(iterations):
carlos.append(monte_carlo_multivariate(price1, price2, days))
a = []
b = []
arr = [0]*days
brr = [0]*days
for carlo in carlos:
i = 0
for carl in carlo:
arr[i] = carl[0]
brr[i] = carl[1]
i+=1
a.append(copy.copy(arr))
b.append(copy.copy(brr))
quant_1 = [0]*days
quant_25 = [0]*days
quant_50 = [0]*days
quant_75 = [0]*days
quant_100 = [0]*days
day_arr = [0]*iterations
for x in range(0,days):
for i in range(0,len(a)-1):
day_arr[i] = a[i][x]
quant_1[x] = np.percentile(day_arr, 1)
quant_25[x] = np.percentile(day_arr, 25)
quant_50[x] = np.percentile(day_arr, 50)
quant_75[x] = np.percentile(day_arr, 75)
quant_100[x] = np.percentile(day_arr, 100)
print(quant_25[-1], quant_50[-1], quant_75[-1])
days_arr = [i for i in range(1,days+1)]
fig, ax = plt.subplots()
plt.plot(quant_50)
ax.fill_between(days_arr, quant_1, quant_100, color='r', alpha=0.4)
ax.fill_between(days_arr, quant_25, quant_75, color='r', alpha=0.7)
plt.show()
for x in range(0,days):
for i in range(0,len(b)-1):
day_arr[i] = b[i][x]
quant_1[x] = np.percentile(day_arr, 1)
quant_25[x] = np.percentile(day_arr, 25)
quant_50[x] = np.percentile(day_arr, 50)
quant_75[x] = np.percentile(day_arr, 75)
quant_100[x] = np.percentile(day_arr, 100)
print(quant_25[-1], quant_50[-1], quant_75[-1])
fig, ax = plt.subplots()
plt.plot(quant_50)
ax.fill_between(days_arr, quant_1, quant_100, color='r', alpha=0.4)
ax.fill_between(days_arr, quant_25, quant_75, color='r', alpha=0.7)
plt.show()
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
fan_chart(prices['price ltc-usd'], prices['price nvidia'], days=30)
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
fan_chart(prices['price btc-usd'], prices['price nvidia'], days=30)
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
fan_chart(prices['price ripple-usd'], prices['price nvidia'], days=30)
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plt.rcParams['figure.figsize'] = (20.0, 10.0)
plt.rcParams['font.size'] = (20.0)
fan_chart(prices['price ripple-usd'], prices['price crude-oil-prices-europe-usd'], days=30)
The copula library we found works partly, there are a couple of issues we needed to handle to get it to work and there's also some little changes to the library itself. The reason is that the author of the library has basically not continued to maintain it and some of the functions depend on a very specific library which is not installable directly in python 3.6 anymore and is a C wrapper basically (http://bonsai.hgc.jp/~mdehoon/software/python/Statistics/manual/index.xhtml). What we did manage is to construct a copula but not calculate the inverse of the cumulative probability distribution.
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def copula_restructure(copula, days):
matrix1 = [0]*days
matrix2 = [0]*days
matrix3 = [0]*days
for i in range(0,len(copula[0])):
matrix1[i] = copula[0][i]
for i in range(0,len(copula[1])):
matrix2[i] = copula[1][i]
for i in range(0, days):
matrix3[i] = [matrix1[i], matrix2[i]]
return matrix3
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plt.rcParams['figure.figsize'] = (5, 4)
from copulalib.copulalib import Copula
foo=Copula(prices['price btc-usd'], prices['price nvidia'], family='clayton')
uc, vc = foo.generate_uv(1000)
plt.scatter(uc, vc, marker='.', color='r')
plt.show()
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plt.rcParams['figure.figsize'] = (20, 10)
from copulalib.copulalib import Copula
foo=Copula(prices['price btc-usd'], prices['price nvidia'], family='clayton')
uc, vc = foo.generate_uv(10000)
plt.scatter(uc, vc, marker='.', color='r')
plt.show()
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def monte_carlo_multivariate_copula(price1, price2, days):
#means = [np.mean(price1.pct_change().loc[1:]), np.mean(price2.pct_change().loc[1:])]
normal_multi=Copula(price1, price2, family='clayton')
normal_multi=normal_multi.generate_uv(days)
normal_multi=np.asmatrix(copula_restructure(normal_multi, days))
for i in range(len(normal_multi)):
normal_multi[i-1] = normal_multi[i-1]+1
A = []
A.append([price1.loc[1399], price2.loc[1399]])
for i in range(1,days):
A.append(np.multiply(A[i-1],normal_multi[i-1]))
return A
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monte_carlo_multivariate_copula(prices['price ltc-usd'], prices['price nvidia'], days=30)
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foo=Copula(prices['price amd-usd'], prices['price btc-usd'], family='clayton')
foo.generate_uv(10)
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#z = zip(prices['price amd-usd'], prices['price btc-usd'])
print(prices['price amd-usd'].pct_change().loc[1:], prices['price ltc-usd'].pct_change().loc[1:])
means = [np.mean(prices['price amd-usd'].pct_change().loc[1:]), np.mean(prices['price ltc-usd'].pct_change().loc[1:])]
cov = np.cov(prices['price amd-usd'].pct_change().loc[1:], prices['price ltc-usd'].pct_change().loc[1:])
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normal_multi = np.random.multivariate_normal(means, cov, 1400)
for i in range(len(normal_multi)):
normal_multi[i-1] = normal_multi[i-1]+1
#normal_multi = normal_multi/normal_multi.sum()
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
#plt.plot(normal_multi)
print(normal_multi)
print((np.std(normal_multi)**2)/2)
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normal_multi.generate_xy()
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A = [0]*100
A[0] = [prices['price amd-usd'].loc[1399], prices['price btc-usd'].loc[1399]]
for i in range(1,30):
A[i] = A[i-1]*normal_multi[i]
plt.plot(A)
print(A[0][0])
#plt.plot(A)
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from copulalib.copulalib import Copula
foo=Copula(prices['price amd-usd'], prices['price btc-usd'], family='clayton')
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uc, vc = foo.generate_uv(1000)
#fig = plt.figure()
#fig.add_subplot(4,2,2)
#plt.scatter(uc, vc, marker='.', color='r')
plt.scatter(uc, vc, marker='.', color='r')
plt.show()
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foo=Copula(prices['price amd-usd'], prices['price btc-usd'],family='frank')
uc, vc = foo.generate_uv(1400)
plt.scatter(uc, vc, marker='.', color='r')
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foo=Copula(prices['price amd-usd'], prices['price ltc-usd'],family='gumbel')
uc, vc = foo.generate_uv(1000)
#fig = plt.figure()
#fig.add_subplot(4,2,2)
plt.scatter(uc, vc, marker='.', color='r')
plt.show()
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print(foo.sr)
plt.plot(prices['price amd-usd'][0:299])
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plt.plot(prices['price nvidia'])
plt.plot(prices['price btc-usd']/15)
plt.show()
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from sklearn.svm import SVR
import matplotlib.pyplot as pyplot
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svr_len=SVR(kernel="linear", C=1e3)
svr_poly = SVR(kernel="polynomial", C=1e3, degree=2)
svr_rbf =SVR(kernel="rbf", C=1e3, gamma=0.1)
svr_lim